I was reading the Cramer Shoup CCA2 secure encryption scheme. The scheme is as follows.
Public key = $(g_1, g_2, c, d, h, hk)$, where $c = g_1^{x_1}g_2^{y_1}$, $d = g_1^{x_2}g_2^{y_2}$, $h = g_1^z$, hk is hash key of a universal one-way hash function.
Secret key = $(x_1, y_1, x_2, y_2, z)$
Encryption(pk, m; r) = $(u_1, u_2, e, v)$, where $u_1 = g_1^r, u_2 = g_2^r, e = h^r.m, \alpha = \mathsf{Hash}(hk, (u_1, u_2, e)), v = c^r.d^{r\alpha}$.
In elgamal scheme, the ciphertext consists of only $u_1, e$ terms. I didn't fully understand the intuition behind the additional terms in the public key and ciphertext. What purpose is each term serving? What terms are eradicating CCA1 attack and what terms are eradicating CCA2 attack?
I came up with this simpler scheme, and couldn't find a CCA attack on this. Is this scheme secure?
Public key = $(g_1, c, d, h, hk)$, where $c = g_1^x$, $d = g_1^y$, $h = g_1^z$, $hk$ is hash key of a universal one-way hash function.
Secret key = $x, y, z$
Encryption(pk, m; r) = $(u_1, e, v)$, where $u_1 = g_1^r, e = h^r.m, \alpha = \mathsf{Hash}(hk, (u_1, e)), v = c^r.d^{r\alpha}$.